2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 351356
author:  Jun Tarui 

title:  On the Minimum Number of Completely 3Scrambling Permutations 
keywords:  
abstract: 
A family
P
= {π
1
,…,π
q
}
[n]={1,…,n}
is completely
k
scrambling [Spencer, 1972;
Füredi, 1996] if for any distinct
k
points
x
, permutations
1
,…,x
k
∈[n]
π
's in
i
P
k!
possible orders on
π
. Let
i
(x
1
),…,π
i
(x
k
)
N
be the minimum size of such a family. This paper
focuses on the case
*
(n,k)
k=3
. By a simple explicit construction, we show the
following upper bound, which we express together with the
lower bound due to Füredi for comparison.
2 /
We also prove the existence of
log
2
e
log
2
n ≤ N
*
(n,3) ≤2
log
2
n + (1+o(1))
log
2
log
2
n.
lim
n→∞
N
*
(n,3) /
log
2
n = c
3
c
and proving the existence of
3
lim
n→∞
N
*
(n,k) /
log
2
n = c
k
k ≥4
remain open.

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reference:  Jun Tarui (2005), On the Minimum Number of Completely 3Scrambling Permutations, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 351356 
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